1 Grouped and Individual Data

“The data were generated from random numbers, and there is no relation between X and Y at all. Firstly, values of X and Y were generated for each ‘subject,’ then a further random number was added to make the individual observation.”

From Bland and Altman, BMJ, 1994, 308, 896.

So… we follow their procedure.

1.1 Grouped Data

1.2 Individual Data

2 Analyses

2.1 OLS

## 
## Call:
## lm(formula = y.new ~ x.new, data = individual_dataset)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -48.787  -7.580   4.742  14.604  33.979 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -4.8731    14.8849  -0.327 0.746339    
## x.new         0.9154     0.2345   3.904 0.000713 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 25.12 on 23 degrees of freedom
## Multiple R-squared:  0.3986, Adjusted R-squared:  0.3724 
## F-statistic: 15.24 on 1 and 23 DF,  p-value: 0.0007132

2.2 MLM

## Linear mixed model fit by REML ['lmerMod']
## Formula: y.new ~ x.new + (1 | groupnum)
##    Data: individual_dataset
## 
## REML criterion at convergence: 182.1
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -1.3098 -0.6503 -0.3808  0.6974  1.5197 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  groupnum (Intercept) 858.47   29.300  
##  Residual              46.83    6.843  
## Number of obs: 25, groups:  groupnum, 5
## 
## Fixed effects:
##             Estimate Std. Error t value
## (Intercept)  27.2568    17.7559   1.535
## x.new         0.3777     0.1992   1.896
## 
## Correlation of Fixed Effects:
##       (Intr)
## x.new -0.670